Question

Evaluate: \( \int e^{x} \sin x \cos x \, dx \)

• A. \( \frac{e^{x} \sin^{2} x}{2} + C \)
• B. \( \frac{e^{x} \cos^{2} x}{2} + C \)
• C. \( \frac{e^{x} \sin 2x}{4} + C \)
• D. \( \frac{e^{x}}{10} (\sin 2x - 2 \cos 2x) + C \)

Hint:

When you see $\sin x \cos x$, always convert it to $\frac{1}{2}\sin 2x$. It reduces two trigonometric terms into one, making the integration much simpler.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
This is an integral involving the product of an exponential and a trigonometric function. First, we simplify the trigonometric part using double-angle identities.

Step 2: Key Formula or Approach
1. Use \( \sin x \cos x = \frac{1}{2} \sin 2x \). 2. Use the standard result: \( \int e^{ax} \sin bx \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx) + C \).

Step 3: Detailed Explanation
1. Simplify the integral: \[ I = \int e^x \left( \frac{1}{2} \sin 2x \right) dx = \frac{1}{2} \int e^x \sin 2x \, dx \] 2. Identify parameters for the standard formula: $a = 1, b = 2$. 3. Apply the formula: \[ I = \frac{1}{2} \left[ \frac{e^x}{1^2 + 2^2} (1 \cdot \sin 2x - 2 \cdot \cos 2x) \right] + C \] \[ I = \frac{1}{2} \left[ \frac{e^x}{5} (\sin 2x - 2 \cos 2x) \right] + C \] \[ I = \frac{e^x}{10} (\sin 2x - 2 \cos 2x) + C \]

Step 4: Final Answer
The value of the integral is \( \frac{e^x}{10} (\sin 2x - 2 \cos 2x) + C \).